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Black Body Radiation: Determination of Stefan's Constant
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Aim: 

 

Determination of Stefan- Boltzmann constant σ .

 

Apparatus:

 

Heater, temperature-indicators, box containing metallic hemisphere with provision for water-flow through its annulus, a suitable black body which can be connected at the bottom of this metallic hemisphere.

 

Principle:

 

A black body is an ideal body which absorbs or emits all types of electromagnetic radiation. The term ‘black body’ was first coined by the German physicist Kirchhoff during 1860’s. Black body radiation is the type of electromagnetic radiation emitted by a black body at constant temperature. The spectrum of this radiation is specific and its intensity depends only on the temperature of the black body. It was the study of this phenomenon which led to a new branch of physics called Quantum mechanics.


According to Stefan’s Boltzmann law (formulated by the Austrian physicists, Stefan and Boltzmann), energy radiated per unit area per unit time by a body is given by,


«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«mo»=«/mo»«mi»§#949;«/mi»«mi»§#963;«/mi»«msup»«mi»T«/mi»«mn»4«/mn»«/msup»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»1«/mn»«mo»)«/mo»«/math»

 

Where R = energy radiated per area per time, Є = emissivity of the material of the body, σ = Stefan’s constant = 5.67x10-8 Wm-2K-4, and T is the temperature in Kelvin scale.

 

 

 

                     Josef Stefan                                                                                                                                    Ludwig Boltzmann 

 

 

For an ideal black body, emissivity Є=1, and equation (1) becomes,

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«mo»=«/mo»«mi»§#963;«/mi»«msup»«mi»T«/mi»«mn»4«/mn»«/msup»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«/math»

                                                                                                                   

The block diagram of experimental set up to study the blackbody radiation is given below.

                                                                                                  figure(1)        

 

                      

This setup uses a copper disc as an approximation to the black body disc which absorbs radiation from the metallic hemisphere as shown in fig (1). Let Td and Th is the steady state temperatures of copper disc and metallic hemisphere respectively. Now according to the equation (2), the net heat transfer to the copper disc per second is,

 

 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»§#916;«/mi»«mi»Q«/mi»«/mrow»«mrow»«mi»§#916;«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»§#963;«/mi»«mi»A«/mi»«mfenced»«mrow»«msup»«msub»«mi»T«/mi»«mi»h«/mi»«/msub»«mn»4«/mn»«/msup»«mo»-«/mo»«msup»«msub»«mi»T«/mi»«mi»d«/mi»«/msub»«mn»4«/mn»«/msup»«/mrow»«/mfenced»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»3«/mn»«mo»)«/mo»«/math»

 

Where A is the area of the copper disc and ΔQ= (Qh-Qd).

 

Now, we have another equation from thermodynamics for heat transfer as,

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»§#916;«/mi»«mi»Q«/mi»«/mrow»«mrow»«mi»§#916;«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»m«/mi»«msub»«mi»C«/mi»«mi»p«/mi»«/msub»«mfrac»«mrow»«mi»d«/mi»«mi»T«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»4«/mn»«mo»)«/mo»«/math»

Where ‘m’ mass of the disc, ‘Cp’’ specific heat of the copper, dT/dt is the change in temperature per unit time.

 

Equating equations (3) and (4),

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#963;«/mi»«mi»A«/mi»«mfenced»«mrow»«msup»«msub»«mi»T«/mi»«mi»h«/mi»«/msub»«mn»4«/mn»«/msup»«mo»-«/mo»«msup»«msub»«mi»T«/mi»«mi»d«/mi»«/msub»«mn»4«/mn»«/msup»«/mrow»«/mfenced»«mo»=«/mo»«mi»m«/mi»«msub»«mi»C«/mi»«mi»p«/mi»«/msub»«mfrac»«mrow»«mi»d«/mi»«mi»T«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»5«/mn»«mo»)«/mo»«/math»

Hence,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#963;«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»m«/mi»«msub»«mi»C«/mi»«mi»p«/mi»«/msub»«/mrow»«mrow»«mi»A«/mi»«mfenced»«mrow»«msup»«msub»«mi»T«/mi»«mi»h«/mi»«/msub»«mn»4«/mn»«/msup»«mo»-«/mo»«msup»«msub»«mi»T«/mi»«mi»d«/mi»«/msub»«mn»4«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/mfrac»«mfrac»«mrow»«mi»d«/mi»«mi»T«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»(«/mo»«mn»6«/mn»«mo»)«/mo»«/math»

 

Applications:

 

  1. Determination of  temperature of Sun from its energy flux density.
  2. Temperature of stars other than Sun, and also their radius relative to the Sun, can be approximated by similar means.
  3. We can find the temperature of Earth, by equating the energy received from the Sun and the energy transmitted by the Earth under black body approximation.

 

 

 

 

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by
Amrita CREATE (Center for Research in Advanced Technologies for Education),
VALUE (Virtual Amrita Laboratories Universalizing Education)
Amrita University, India 2009 - 2014
http://www.amrita.edu/create

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